How to model Harry Potter's Firebolt?
Use Ctrl+R to loop cut. This is useful for making new edges/verts around an object. Do this and select the part of the handle, then press S to scale everything up so it matches. All in Edit Mode of course.

Additional tip:
For the tip of the broom, you can either make "hair" strands (Look up hair/grass/fur tutorials), or you can extrude the broom staff all the way back, scale it up a bit, and add transparent textures with the strands. The last one is most useful if you're going to use this in a game though.

You can also subdivide your object and enter sculpt mode to make the patterns on the broom handle.